A000702 a(n) is the number of conjugacy classes in the alternating group A_n. (Formerly M2307 N0910)

1, 1, 3, 4, 5, 7, 9, 14, 18, 24, 31, 43, 55, 72, 94, 123, 156, 200, 254, 324, 408, 513, 641, 804, 997, 1236, 1526, 1883, 2308, 2829, 3451, 4209, 5109, 6194, 7485, 9038, 10871, 13063, 15654, 18738, 22365, 26665, 31716, 37682, 44669, 52887, 62494, 73767

Offset

1,3

References

Girse, Robert D.; The number of conjugacy classes of the alternating group. Nordisk Tidskr. Informationsbehandling (BIT) 20 (1980), no. 4, 515-517.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000 [a(1)=1 prepended by Georg Fischer, Sep 29 2020]

R. D. Girse, The number of conjugacy classes of the alternating group, Preprint and correspondence [Annotated scanned copy]

M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

Eric Weisstein's World of Mathematics, Alternating Group.

Index entries for sequences related to groups

Formula

a(n) = (p(n) + 3Q(n))/2 for n>1 where p(n) denotes the number of unrestricted partitions of n (A000041) and Q(n) the number of partitions of n into distinct odd parts (A000700). [Denes-Erdős-Turan]

a(n) = 2p(n) + 3*Sum_{r>=1} (-1)^r*p(n-2r^2) for n>1. [Girse]

Sum_{r>=0} (-1)^r*a(n-(3r^2 +- r)/2) = 3(-1)^t if n = 2t^2 or 0 otherwise, where p(u) and a(u) are taken as 0 unless u is a nonnegative integer and t = 1,2,3,... [Girse]

Example

G.f. = x + x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 14*x^8 + 18*x^9 + ...

Mathematica

p = PartitionsP; q[n_] := SeriesCoefficient[ Product[ 1+x^(2k+1), {k, 0, n}], {x, 0, n}]; a[1]=1; a[n_] := (p[n] + 3*q[n])/2; Table[a[n], {n, 48}] (* Jean-François Alcover, Feb 22 2012, after first formula *)
a[ n_] := SeriesCoefficient[ ( 1 / QPochhammer[ x] + 3 / QPochhammer[ x, -x] ) / 2 - (2 + x), {x, 0, n}]; Table[a[n], {n, 1, 48}] (* Michael Somos, May 28 2014 *)

Prog

(Magma) [ NumberOfClasses(Alt(n)) : n in [1..10] ]; /* A useful example of MAGMA code, but it is better to use one of the formulas as below: */ A000702:= func< n | 2*NumberOfPartitions(n)+3*(&+[(-1)^r*NumberOfPartitions(n-2*r^2): r in [1..Isqrt(n div 2)]]) >; [1] cat [A000702(n): n in [2..48]]; // Jason Kimberley, Feb 01 2011
(PARI) default(seriesprecision, 99);
Vec((1/eta(x)+3*eta(x^2)^2/(eta(x)*eta(x^4)))/2-(2+x)) /* Joerg Arndt, Feb 02 2011 */

Crossrefs

Cf. A073584.

Sequence in context: A259796 A082922 A036971 * A067526 A101760 A165713

Adjacent sequences: A000699 A000700 A000701 * A000703 A000704 A000705

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

a(1)=1 added by N. J. A. Sloane, Sep 14 2020

Follow-up corrections by Andrey Zabolotskiy, Sep 18 2020

Status

approved